Tomography, the computation (estimation) of a density in a region in a n-dimensional space based on m-dimensional projections of that region (represented as pixels; usually 0<m<n), falls into two major categories: filtered back-projection (FBP) and iterative reconstruction (IR), which is also known as a variant of the Algebraic Reconstruction Technique (ART). Both FBP and conventional IR techniques have specific deficiencies, especially in modern settings that often require 106-1010 image elements (i.e., voxels) to be computed when one projection often contains anywhere from 2000 pixels (one-dimensional) up to 2000×2000 pixels (two-dimensional).
Major deficiencies associated with FBP include the need for a large number of projections to achieve limited quantitative accuracy. The number of projections is typically counted in the hundreds but the projections are not used as efficiently as they might. For example, computed density estimates may be negative while it is known that physical intensities cannot be negative. The limited quantitative accuracy may sometimes be improved by subsequent iterative refinement. Other deficiencies associated with the FBP include the inability to change data-weights depending on voxel location, as discussed in Wood, S. L., Morf, M., A fast implementation of a minimum variance estimator for computerized tomography image reconstruction, IEEE, Trans. on Biomed. Eng., Vol. BME-28, No. 2, pp. 56-68, 1981 (hereinafter Wood) and the inability to effectively consider a variety of constraints.
The strength of the FBP method include its computation speed as it typically relies on the fast Fourier transformation, central slice theorem, and possibly the use of suitable and pre-computed filters.
Alternative techniques to the FBP and conventional IR techniques use matrix operations and may be applicable for “small” problems in which the number of volume or picture elements (voxels or pixels) is in the few-thousands. For typical tomographic reconstruction settings, however, the computational burden cannot be handled in the foreseeable future by these matrix-based techniques. This is because the number of computer operations scales as a power of N. For example, in a 3-D case the operations count could scale faster than N5 (where N is the number of pixels of a side of a reconstruction cube, representing the region of interest), as discussed in Wood and U.S. Pat. No. 6,332,035, Dec. 18, 2001.
The benefits of IR techniques include their ability to account for constraints, especially the ability to assure density estimates to be non-negative. Assuring non-negative density estimates may lead to a significant image contrast improvement and better quantitative results.
The deficiencies associated with conventional IR techniques include the need to solve repeatedly an inversion operation as well as a forward projection operation in order to obtain corrections terms for subsequent iterations. The concomitant need for many recursive pairings of these operations usually make the IR techniques slow and, depending on particular implementations, unstable.
Because these conventional IR techniques use various (iterative) optimization methods, their rate of convergence to the solution depends on how well a particular technique accounts for the Hessian matrix, which is the second derivative matrix of the objective function to be optimized (i.e., the performance criterion relative to image elements). The most popular techniques are variants of IR. Due to the large size of the Hessian matrix for image reconstruction, however, its structure (and that of its inverse) are typically ignored or poorly approximated. Furthermore, because of the wide distribution of eigen-values of the Hessian, current optimization techniques tend to show no improvement beyond a number of iterations (typically counted in the tens-to-hundreds) and may only cope with the large eigenvalues.
Multi-grid variations of these algorithms may help, but ultimately still fail because of the size of the Hessians involved with fine grids. Multi-grid resolution here refers to the use of progressively finer resolution as iterations are performed.